On the Interplay Between Probability Distributions, Graphs, and Matrices

• Author(s): Nancy Wermuth*+
• Companies: Chalmers/Gothenburg University

Abstract

The independence structure of a joint probability distribution may be captured by a graph. Such graphs consist of nodes for random variables and of edges that couple node pairs. They differ by the type and number of edges present. Questions concerning independence structures arise in particular when different studies are to be compared. Examples are to decide whether two different types of graph capture the same structure or how a given structure changes for a subset of variables and for given levels of some variables. One road to answer such questions is to represent different types of edge by corresponding binary matrices and to develop operators for these binary matrices that lead to repeated transformations of graphs. Thereby, the independence structure in an associated derived probability distribution becomes largely a consequence of the properties of the matrix operators.